(* Title: HOL/Analysis/Measure_Space.thy
Author: Lawrence C Paulson
Author: Johannes Hölzl, TU München
Author: Armin Heller, TU München
*)
section \<open>Measure Spaces\<close>
theory Measure_Space
imports
Measurable "HOL-Library.Extended_Nonnegative_Real"
begin
subsection\<^marker>\<open>tag unimportant\<close> "Relate extended reals and the indicator function"
lemma suminf_cmult_indicator:
fixes f :: "nat \ ennreal"
assumes "disjoint_family A" "x \ A i"
shows "(\n. f n * indicator (A n) x) = f i"
proof -
have **: "\n. f n * indicator (A n) x = (if n = i then f n else 0 :: ennreal)"
using \<open>x \<in> A i\<close> assms unfolding disjoint_family_on_def indicator_def by auto
then have "\n. (\j
by (auto simp: sum.If_cases)
moreover have "(SUP n. if i < n then f i else 0) = (f i :: ennreal)"
proof (rule SUP_eqI)
fix y :: ennreal assume "\n. n \ UNIV \ (if i < n then f i else 0) \ y"
from this[of "Suc i"] show "f i \ y" by auto
qed (insert assms, simp)
ultimately show ?thesis using assms
by (subst suminf_eq_SUP) (auto simp: indicator_def)
qed
lemma suminf_indicator:
assumes "disjoint_family A"
shows "(\n. indicator (A n) x :: ennreal) = indicator (\i. A i) x"
proof cases
assume *: "x \ (\i. A i)"
then obtain i where "x \ A i" by auto
from suminf_cmult_indicator[OF assms(1), OF \<open>x \<in> A i\<close>, of "\<lambda>k. 1"]
show ?thesis using * by simp
qed simp
lemma sum_indicator_disjoint_family:
fixes f :: "'d \ 'e::semiring_1"
assumes d: "disjoint_family_on A P" and "x \ A j" and "finite P" and "j \ P"
shows "(\i\P. f i * indicator (A i) x) = f j"
proof -
have "P \ {i. x \ A i} = {j}"
using d \<open>x \<in> A j\<close> \<open>j \<in> P\<close> unfolding disjoint_family_on_def
by auto
thus ?thesis
unfolding indicator_def
by (simp add: if_distrib sum.If_cases[OF \<open>finite P\<close>])
qed
text \<open>
The type for emeasure spaces is already defined in \<^theory>\<open>HOL-Analysis.Sigma_Algebra\<close>, as it
is also used to represent sigma algebras (with an arbitrary emeasure).
\<close>
subsection\<^marker>\<open>tag unimportant\<close> "Extend binary sets"
lemma LIMSEQ_binaryset:
assumes f: "f {} = 0"
shows "(\n. \i f A + f B"
proof -
have "(\n. \i < Suc (Suc n). f (binaryset A B i)) = (\n. f A + f B)"
proof
fix n
show "(\i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
by (induct n) (auto simp add: binaryset_def f)
qed
moreover
have "... \ f A + f B" by (rule tendsto_const)
ultimately
have "(\n. \i< Suc (Suc n). f (binaryset A B i)) \ f A + f B"
by metis
hence "(\n. \i< n+2. f (binaryset A B i)) \ f A + f B"
by simp
thus ?thesis by (rule LIMSEQ_offset [where k=2])
qed
lemma binaryset_sums:
assumes f: "f {} = 0"
shows "(\n. f (binaryset A B n)) sums (f A + f B)"
by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
lemma suminf_binaryset_eq:
fixes f :: "'a set \ 'b::{comm_monoid_add, t2_space}"
shows "f {} = 0 \ (\n. f (binaryset A B n)) = f A + f B"
by (metis binaryset_sums sums_unique)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Properties of a premeasure \<^term>\<open>\<mu>\<close>\<close>
text \<open>
The definitions for \<^const>\<open>positive\<close> and \<^const>\<open>countably_additive\<close> should be here, by they are
necessary to define \<^typ>\<open>'a measure\<close> in \<^theory>\<open>HOL-Analysis.Sigma_Algebra\<close>.
\<close>
definition subadditive where
"subadditive M f \ (\x\M. \y\M. x \ y = {} \ f (x \ y) \ f x + f y)"
lemma subadditiveD: "subadditive M f \ x \ y = {} \ x \ M \ y \ M \ f (x \ y) \ f x + f y"
by (auto simp add: subadditive_def)
definition countably_subadditive where
"countably_subadditive M f \
(\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
lemma (in ring_of_sets) countably_subadditive_subadditive:
fixes f :: "'a set \ ennreal"
assumes f: "positive M f" and cs: "countably_subadditive M f"
shows "subadditive M f"
proof (auto simp add: subadditive_def)
fix x y
assume x: "x \ M" and y: "y \ M" and "x \ y = {}"
hence "disjoint_family (binaryset x y)"
by (auto simp add: disjoint_family_on_def binaryset_def)
hence "range (binaryset x y) \ M \
(\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"
using cs by (auto simp add: countably_subadditive_def)
hence "{x,y,{}} \ M \ x \ y \ M \
f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
by (simp add: range_binaryset_eq UN_binaryset_eq)
thus "f (x \ y) \ f x + f y" using f x y
by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
qed
definition additive where
"additive M \ \ (\x\M. \y\M. x \ y = {} \ \ (x \ y) = \ x + \ y)"
definition increasing where
"increasing M \ \ (\x\M. \y\M. x \ y \ \ x \ \ y)"
lemma positiveD1: "positive M f \ f {} = 0" by (auto simp: positive_def)
lemma positiveD_empty:
"positive M f \ f {} = 0"
by (auto simp add: positive_def)
lemma additiveD:
"additive M f \ x \ y = {} \ x \ M \ y \ M \ f (x \ y) = f x + f y"
by (auto simp add: additive_def)
lemma increasingD:
"increasing M f \ x \ y \ x\M \ y\M \ f x \ f y"
by (auto simp add: increasing_def)
lemma countably_additiveI[case_names countably]:
"(\A. range A \ M \ disjoint_family A \ (\i. A i) \ M \ (\i. f (A i)) = f (\i. A i))
\<Longrightarrow> countably_additive M f"
by (simp add: countably_additive_def)
lemma (in ring_of_sets) disjointed_additive:
assumes f: "positive M f" "additive M f" and A: "range A \ M" "incseq A"
shows "(\i\n. f (disjointed A i)) = f (A n)"
proof (induct n)
case (Suc n)
then have "(\i\Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
by simp
also have "\ = f (A n \ disjointed A (Suc n))"
using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_mono)
also have "A n \ disjointed A (Suc n) = A (Suc n)"
using \<open>incseq A\<close> by (auto dest: incseq_SucD simp: disjointed_mono)
finally show ?case .
qed simp
lemma (in ring_of_sets) additive_sum:
fixes A:: "'i \ 'a set"
assumes f: "positive M f" and ad: "additive M f" and "finite S"
and A: "A`S \ M"
and disj: "disjoint_family_on A S"
shows "(\i\S. f (A i)) = f (\i\S. A i)"
using \<open>finite S\<close> disj A
proof induct
case empty show ?case using f by (simp add: positive_def)
next
case (insert s S)
then have "A s \ (\i\S. A i) = {}"
by (auto simp add: disjoint_family_on_def neq_iff)
moreover
have "A s \ M" using insert by blast
moreover have "(\i\S. A i) \ M"
using insert \<open>finite S\<close> by auto
ultimately have "f (A s \ (\i\S. A i)) = f (A s) + f(\i\S. A i)"
using ad UNION_in_sets A by (auto simp add: additive_def)
with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
by (auto simp add: additive_def subset_insertI)
qed
lemma (in ring_of_sets) additive_increasing:
fixes f :: "'a set \ ennreal"
assumes posf: "positive M f" and addf: "additive M f"
shows "increasing M f"
proof (auto simp add: increasing_def)
fix x y
assume xy: "x \ M" "y \ M" "x \ y"
then have "y - x \ M" by auto
then have "f x + 0 \ f x + f (y-x)" by (intro add_left_mono zero_le)
also have "... = f (x \ (y-x))" using addf
by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
also have "... = f y"
by (metis Un_Diff_cancel Un_absorb1 xy(3))
finally show "f x \ f y" by simp
qed
lemma (in ring_of_sets) subadditive:
fixes f :: "'a set \ ennreal"
assumes f: "positive M f" "additive M f" and A: "A`S \ M" and S: "finite S"
shows "f (\i\S. A i) \ (\i\S. f (A i))"
using S A
proof (induct S)
case empty thus ?case using f by (auto simp: positive_def)
next
case (insert x F)
hence in_M: "A x \ M" "(\i\F. A i) \ M" "(\i\F. A i) - A x \ M" using A by force+
have subs: "(\i\F. A i) - A x \ (\i\F. A i)" by auto
have "(\i\(insert x F). A i) = A x \ ((\i\F. A i) - A x)" by auto
hence "f (\i\(insert x F). A i) = f (A x \ ((\i\F. A i) - A x))"
by simp
also have "\ = f (A x) + f ((\i\F. A i) - A x)"
using f(2) by (rule additiveD) (insert in_M, auto)
also have "\ \ f (A x) + f (\i\F. A i)"
using additive_increasing[OF f] in_M subs by (auto simp: increasing_def intro: add_left_mono)
also have "\ \ f (A x) + (\i\F. f (A i))" using insert by (auto intro: add_left_mono)
finally show "f (\i\(insert x F). A i) \ (\i\(insert x F). f (A i))" using insert by simp
qed
lemma (in ring_of_sets) countably_additive_additive:
fixes f :: "'a set \ ennreal"
assumes posf: "positive M f" and ca: "countably_additive M f"
shows "additive M f"
proof (auto simp add: additive_def)
fix x y
assume x: "x \ M" and y: "y \ M" and "x \ y = {}"
hence "disjoint_family (binaryset x y)"
by (auto simp add: disjoint_family_on_def binaryset_def)
hence "range (binaryset x y) \ M \
(\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
using ca
by (simp add: countably_additive_def)
hence "{x,y,{}} \ M \ x \ y \ M \
f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
by (simp add: range_binaryset_eq UN_binaryset_eq)
thus "f (x \ y) = f x + f y" using posf x y
by (auto simp add: Un suminf_binaryset_eq positive_def)
qed
lemma (in algebra) increasing_additive_bound:
fixes A:: "nat \ 'a set" and f :: "'a set \ ennreal"
assumes f: "positive M f" and ad: "additive M f"
and inc: "increasing M f"
and A: "range A \ M"
and disj: "disjoint_family A"
shows "(\i. f (A i)) \ f \"
proof (safe intro!: suminf_le_const)
fix N
note disj_N = disjoint_family_on_mono[OF _ disj, of "{..]
have "(\ii\{..
using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N)
also have "... \ f \" using space_closed A
by (intro increasingD[OF inc] finite_UN) auto
finally show "(\i f \" by simp
qed (insert f A, auto simp: positive_def)
lemma (in ring_of_sets) countably_additiveI_finite:
fixes \<mu> :: "'a set \<Rightarrow> ennreal"
assumes "finite \" "positive M \" "additive M \"
shows "countably_additive M \"
proof (rule countably_additiveI)
fix F :: "nat \ 'a set" assume F: "range F \ M" "(\i. F i) \ M" and disj: "disjoint_family F"
have "\i\{i. F i \ {}}. \x. x \ F i" by auto
from bchoice[OF this] obtain f where f: "\i. F i \ {} \ f i \ F i" by auto
have inj_f: "inj_on f {i. F i \ {}}"
proof (rule inj_onI, simp)
fix i j a b assume *: "f i = f j" "F i \ {}" "F j \ {}"
then have "f i \ F i" "f j \ F j" using f by force+
with disj * show "i = j" by (auto simp: disjoint_family_on_def)
qed
have "finite (\i. F i)"
by (metis F(2) assms(1) infinite_super sets_into_space)
have F_subset: "{i. \ (F i) \ 0} \ {i. F i \ {}}"
by (auto simp: positiveD_empty[OF \<open>positive M \<mu>\<close>])
moreover have fin_not_empty: "finite {i. F i \ {}}"
proof (rule finite_imageD)
from f have "f`{i. F i \ {}} \ (\i. F i)" by auto
then show "finite (f`{i. F i \ {}})"
by (rule finite_subset) fact
qed fact
ultimately have fin_not_0: "finite {i. \ (F i) \ 0}"
by (rule finite_subset)
have disj_not_empty: "disjoint_family_on F {i. F i \ {}}"
using disj by (auto simp: disjoint_family_on_def)
from fin_not_0 have "(\i. \ (F i)) = (\i | \ (F i) \ 0. \ (F i))"
by (rule suminf_finite) auto
also have "\ = (\i | F i \ {}. \ (F i))"
using fin_not_empty F_subset by (rule sum.mono_neutral_left) auto
also have "\ = \ (\i\{i. F i \ {}}. F i)"
using \<open>positive M \<mu>\<close> \<open>additive M \<mu>\<close> fin_not_empty disj_not_empty F by (intro additive_sum) auto
also have "\ = \ (\i. F i)"
by (rule arg_cong[where f=\<mu>]) auto
finally show "(\i. \ (F i)) = \ (\i. F i)" .
qed
lemma (in ring_of_sets) countably_additive_iff_continuous_from_below:
fixes f :: "'a set \ ennreal"
assumes f: "positive M f" "additive M f"
shows "countably_additive M f \
(\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i))"
unfolding countably_additive_def
proof safe
assume count_sum: "\A. range A \ M \ disjoint_family A \ \(A ` UNIV) \ M \ (\i. f (A i)) = f (\(A ` UNIV))"
fix A :: "nat \ 'a set" assume A: "range A \ M" "incseq A" "(\i. A i) \ M"
then have dA: "range (disjointed A) \ M" by (auto simp: range_disjointed_sets)
with count_sum[THEN spec, of "disjointed A"] A(3)
have f_UN: "(\i. f (disjointed A i)) = f (\i. A i)"
by (auto simp: UN_disjointed_eq disjoint_family_disjointed)
moreover have "(\n. (\i (\i. f (disjointed A i))"
using f(1)[unfolded positive_def] dA
by (auto intro!: summable_LIMSEQ)
from LIMSEQ_Suc[OF this]
have "(\n. (\i\n. f (disjointed A i))) \ (\i. f (disjointed A i))"
unfolding lessThan_Suc_atMost .
moreover have "\n. (\i\n. f (disjointed A i)) = f (A n)"
using disjointed_additive[OF f A(1,2)] .
ultimately show "(\i. f (A i)) \ f (\i. A i)" by simp
next
assume cont: "\A. range A \ M \ incseq A \ (\i. A i) \ M \ (\i. f (A i)) \ f (\i. A i)"
fix A :: "nat \ 'a set" assume A: "range A \ M" "disjoint_family A" "(\i. A i) \ M"
have *: "(\n. (\ii. A i)" by auto
have "(\n. f (\i f (\i. A i)"
proof (unfold *[symmetric], intro cont[rule_format])
show "range (\i. \i M" "(\i. \i M"
using A * by auto
qed (force intro!: incseq_SucI)
moreover have "\n. f (\ii
using A
by (intro additive_sum[OF f, of _ A, symmetric])
(auto intro: disjoint_family_on_mono[where B=UNIV])
ultimately
have "(\i. f (A i)) sums f (\i. A i)"
unfolding sums_def by simp
from sums_unique[OF this]
show "(\i. f (A i)) = f (\i. A i)" by simp
qed
lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:
fixes f :: "'a set \ ennreal"
assumes f: "positive M f" "additive M f"
shows "(\A. range A \ M \ decseq A \ (\i. A i) \ M \ (\i. f (A i) \ \) \ (\i. f (A i)) \ f (\i. A i))
\<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0)"
proof safe
assume cont: "(\A. range A \ M \ decseq A \ (\i. A i) \ M \ (\i. f (A i) \ \) \ (\i. f (A i)) \ f (\i. A i))"
fix A :: "nat \ 'a set" assume A: "range A \ M" "decseq A" "(\i. A i) = {}" "\i. f (A i) \ \"
with cont[THEN spec, of A] show "(\i. f (A i)) \ 0"
using \<open>positive M f\<close>[unfolded positive_def] by auto
next
assume cont: "\A. range A \ M \ decseq A \ (\i. A i) = {} \ (\i. f (A i) \ \) \ (\i. f (A i)) \ 0"
fix A :: "nat \ 'a set" assume A: "range A \ M" "decseq A" "(\i. A i) \ M" "\i. f (A i) \ \"
have f_mono: "\a b. a \ M \ b \ M \ a \ b \ f a \ f b"
using additive_increasing[OF f] unfolding increasing_def by simp
have decseq_fA: "decseq (\i. f (A i))"
using A by (auto simp: decseq_def intro!: f_mono)
have decseq: "decseq (\i. A i - (\i. A i))"
using A by (auto simp: decseq_def)
then have decseq_f: "decseq (\i. f (A i - (\i. A i)))"
using A unfolding decseq_def by (auto intro!: f_mono Diff)
have "f (\x. A x) \ f (A 0)"
using A by (auto intro!: f_mono)
then have f_Int_fin: "f (\x. A x) \ \"
using A by (auto simp: top_unique)
{ fix i
have "f (A i - (\i. A i)) \ f (A i)" using A by (auto intro!: f_mono)
then have "f (A i - (\i. A i)) \ \"
using A by (auto simp: top_unique) }
note f_fin = this
have "(\i. f (A i - (\i. A i))) \ 0"
proof (intro cont[rule_format, OF _ decseq _ f_fin])
show "range (\i. A i - (\i. A i)) \ M" "(\i. A i - (\i. A i)) = {}"
using A by auto
qed
from INF_Lim[OF decseq_f this]
have "(INF n. f (A n - (\i. A i))) = 0" .
moreover have "(INF n. f (\i. A i)) = f (\i. A i)"
by auto
ultimately have "(INF n. f (A n - (\i. A i)) + f (\i. A i)) = 0 + f (\i. A i)"
using A(4) f_fin f_Int_fin
by (subst INF_ennreal_add_const) (auto simp: decseq_f)
moreover {
fix n
have "f (A n - (\i. A i)) + f (\i. A i) = f ((A n - (\i. A i)) \ (\i. A i))"
using A by (subst f(2)[THEN additiveD]) auto
also have "(A n - (\i. A i)) \ (\i. A i) = A n"
by auto
finally have "f (A n - (\i. A i)) + f (\i. A i) = f (A n)" . }
ultimately have "(INF n. f (A n)) = f (\i. A i)"
by simp
with LIMSEQ_INF[OF decseq_fA]
show "(\i. f (A i)) \ f (\i. A i)" by simp
qed
lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:
fixes f :: "'a set \ ennreal"
assumes f: "positive M f" "additive M f" "\A\M. f A \ \"
assumes cont: "\A. range A \ M \ decseq A \ (\i. A i) = {} \ (\i. f (A i)) \ 0"
assumes A: "range A \ M" "incseq A" "(\i. A i) \ M"
shows "(\i. f (A i)) \ f (\i. A i)"
proof -
from A have "(\i. f ((\i. A i) - A i)) \ 0"
by (intro cont[rule_format]) (auto simp: decseq_def incseq_def)
moreover
{ fix i
have "f ((\i. A i) - A i \ A i) = f ((\i. A i) - A i) + f (A i)"
using A by (intro f(2)[THEN additiveD]) auto
also have "((\i. A i) - A i) \ A i = (\i. A i)"
by auto
finally have "f ((\i. A i) - A i) = f (\i. A i) - f (A i)"
using f(3)[rule_format, of "A i"] A by (auto simp: ennreal_add_diff_cancel subset_eq) }
moreover have "\\<^sub>F i in sequentially. f (A i) \ f (\i. A i)"
using increasingD[OF additive_increasing[OF f(1, 2)], of "A _" "\i. A i"] A
by (auto intro!: always_eventually simp: subset_eq)
ultimately show "(\i. f (A i)) \ f (\i. A i)"
by (auto intro: ennreal_tendsto_const_minus)
qed
lemma (in ring_of_sets) empty_continuous_imp_countably_additive:
fixes f :: "'a set \ ennreal"
assumes f: "positive M f" "additive M f" and fin: "\A\M. f A \ \"
assumes cont: "\A. range A \ M \ decseq A \ (\i. A i) = {} \ (\i. f (A i)) \ 0"
shows "countably_additive M f"
using countably_additive_iff_continuous_from_below[OF f]
using empty_continuous_imp_continuous_from_below[OF f fin] cont
by blast
subsection\<^marker>\<open>tag unimportant\<close> \<open>Properties of \<^const>\<open>emeasure\<close>\<close>
lemma emeasure_positive: "positive (sets M) (emeasure M)"
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"
using emeasure_positive[of M] by (simp add: positive_def)
lemma emeasure_single_in_space: "emeasure M {x} \ 0 \ x \ space M"
using emeasure_notin_sets[of "{x}" M] by (auto dest: sets.sets_into_space zero_less_iff_neq_zero[THEN iffD2])
lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
lemma suminf_emeasure:
"range A \ sets M \ disjoint_family A \ (\i. emeasure M (A i)) = emeasure M (\i. A i)"
using sets.countable_UN[of A UNIV M] emeasure_countably_additive[of M]
by (simp add: countably_additive_def)
lemma sums_emeasure:
"disjoint_family F \ (\i. F i \ sets M) \ (\i. emeasure M (F i)) sums emeasure M (\i. F i)"
unfolding sums_iff by (intro conjI suminf_emeasure) auto
lemma emeasure_additive: "additive (sets M) (emeasure M)"
by (metis sets.countably_additive_additive emeasure_positive emeasure_countably_additive)
lemma plus_emeasure:
"a \ sets M \ b \ sets M \ a \ b = {} \ emeasure M a + emeasure M b = emeasure M (a \ b)"
using additiveD[OF emeasure_additive] ..
lemma emeasure_Un:
"A \ sets M \ B \ sets M \ emeasure M (A \ B) = emeasure M A + emeasure M (B - A)"
using plus_emeasure[of A M "B - A"] by auto
lemma emeasure_Un_Int:
assumes "A \ sets M" "B \ sets M"
shows "emeasure M A + emeasure M B = emeasure M (A \ B) + emeasure M (A \ B)"
proof -
have "A = (A-B) \ (A \ B)" by auto
then have "emeasure M A = emeasure M (A-B) + emeasure M (A \ B)"
by (metis Diff_Diff_Int Diff_disjoint assms plus_emeasure sets.Diff)
moreover have "A \ B = (A-B) \ B" by auto
then have "emeasure M (A \ B) = emeasure M (A-B) + emeasure M B"
by (metis Diff_disjoint Int_commute assms plus_emeasure sets.Diff)
ultimately show ?thesis by (metis add.assoc add.commute)
qed
lemma sum_emeasure:
"F`I \ sets M \ disjoint_family_on F I \ finite I \
(\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
by (metis sets.additive_sum emeasure_positive emeasure_additive)
lemma emeasure_mono:
"a \ b \ b \ sets M \ emeasure M a \ emeasure M b"
by (metis zero_le sets.additive_increasing emeasure_additive emeasure_notin_sets emeasure_positive increasingD)
lemma emeasure_space:
"emeasure M A \ emeasure M (space M)"
by (metis emeasure_mono emeasure_notin_sets sets.sets_into_space sets.top zero_le)
lemma emeasure_Diff:
assumes finite: "emeasure M B \ \"
and [measurable]: "A \ sets M" "B \ sets M" and "B \ A"
shows "emeasure M (A - B) = emeasure M A - emeasure M B"
proof -
have "(A - B) \ B = A" using \B \ A\ by auto
then have "emeasure M A = emeasure M ((A - B) \ B)" by simp
also have "\ = emeasure M (A - B) + emeasure M B"
by (subst plus_emeasure[symmetric]) auto
finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
using finite by simp
qed
lemma emeasure_compl:
"s \ sets M \ emeasure M s \ \ \ emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
by (rule emeasure_Diff) (auto dest: sets.sets_into_space)
lemma Lim_emeasure_incseq:
"range A \ sets M \ incseq A \ (\i. (emeasure M (A i))) \ emeasure M (\i. A i)"
using emeasure_countably_additive
by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive
emeasure_additive)
lemma incseq_emeasure:
assumes "range B \ sets M" "incseq B"
shows "incseq (\i. emeasure M (B i))"
using assms by (auto simp: incseq_def intro!: emeasure_mono)
lemma SUP_emeasure_incseq:
assumes A: "range A \ sets M" "incseq A"
shows "(SUP n. emeasure M (A n)) = emeasure M (\i. A i)"
using LIMSEQ_SUP[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A]
by (simp add: LIMSEQ_unique)
lemma decseq_emeasure:
assumes "range B \ sets M" "decseq B"
shows "decseq (\i. emeasure M (B i))"
using assms by (auto simp: decseq_def intro!: emeasure_mono)
lemma INF_emeasure_decseq:
assumes A: "range A \ sets M" and "decseq A"
and finite: "\i. emeasure M (A i) \ \"
shows "(INF n. emeasure M (A n)) = emeasure M (\i. A i)"
proof -
have le_MI: "emeasure M (\i. A i) \ emeasure M (A 0)"
using A by (auto intro!: emeasure_mono)
hence *: "emeasure M (\i. A i) \ \" using finite[of 0] by (auto simp: top_unique)
have "emeasure M (A 0) - (INF n. emeasure M (A n)) = (SUP n. emeasure M (A 0) - emeasure M (A n))"
by (simp add: ennreal_INF_const_minus)
also have "\ = (SUP n. emeasure M (A 0 - A n))"
using A finite \<open>decseq A\<close>[unfolded decseq_def] by (subst emeasure_Diff) auto
also have "\ = emeasure M (\i. A 0 - A i)"
proof (rule SUP_emeasure_incseq)
show "range (\n. A 0 - A n) \ sets M"
using A by auto
show "incseq (\n. A 0 - A n)"
using \<open>decseq A\<close> by (auto simp add: incseq_def decseq_def)
qed
also have "\ = emeasure M (A 0) - emeasure M (\i. A i)"
using A finite * by (simp, subst emeasure_Diff) auto
finally show ?thesis
by (rule ennreal_minus_cancel[rotated 3])
(insert finite A, auto intro: INF_lower emeasure_mono)
qed
lemma INF_emeasure_decseq':
assumes A: "\i. A i \ sets M" and "decseq A"
and finite: "\i. emeasure M (A i) \ \"
shows "(INF n. emeasure M (A n)) = emeasure M (\i. A i)"
proof -
from finite obtain i where i: "emeasure M (A i) < \"
by (auto simp: less_top)
have fin: "i \ j \ emeasure M (A j) < \" for j
by (rule le_less_trans[OF emeasure_mono i]) (auto intro!: decseqD[OF \<open>decseq A\<close>] A)
have "(INF n. emeasure M (A n)) = (INF n. emeasure M (A (n + i)))"
proof (rule INF_eq)
show "\j\UNIV. emeasure M (A (j + i)) \ emeasure M (A i')" for i'
by (intro bexI[of _ i'] emeasure_mono decseqD[OF \decseq A\] A) auto
qed auto
also have "\ = emeasure M (INF n. (A (n + i)))"
using A \<open>decseq A\<close> fin by (intro INF_emeasure_decseq) (auto simp: decseq_def less_top)
also have "(INF n. (A (n + i))) = (INF n. A n)"
by (meson INF_eq UNIV_I assms(2) decseqD le_add1)
finally show ?thesis .
qed
lemma emeasure_INT_decseq_subset:
fixes F :: "nat \ 'a set"
assumes I: "I \ {}" and F: "\i j. i \ I \ j \ I \ i \ j \ F j \ F i"
assumes F_sets[measurable]: "\i. i \ I \ F i \ sets M"
and fin: "\i. i \ I \ emeasure M (F i) \ \"
shows "emeasure M (\i\I. F i) = (INF i\I. emeasure M (F i))"
proof cases
assume "finite I"
have "(\i\I. F i) = F (Max I)"
using I \<open>finite I\<close> by (intro antisym INF_lower INF_greatest F) auto
moreover have "(INF i\I. emeasure M (F i)) = emeasure M (F (Max I))"
using I \<open>finite I\<close> by (intro antisym INF_lower INF_greatest F emeasure_mono) auto
ultimately show ?thesis
by simp
next
assume "infinite I"
define L where "L n = (LEAST i. i \ I \ i \ n)" for n
have L: "L n \ I \ n \ L n" for n
unfolding L_def
proof (rule LeastI_ex)
show "\x. x \ I \ n \ x"
using \<open>infinite I\<close> finite_subset[of I "{..< n}"]
by (rule_tac ccontr) (auto simp: not_le)
qed
have L_eq[simp]: "i \ I \ L i = i" for i
unfolding L_def by (intro Least_equality) auto
have L_mono: "i \ j \ L i \ L j" for i j
using L[of j] unfolding L_def by (intro Least_le) (auto simp: L_def)
have "emeasure M (\i. F (L i)) = (INF i. emeasure M (F (L i)))"
proof (intro INF_emeasure_decseq[symmetric])
show "decseq (\i. F (L i))"
using L by (intro antimonoI F L_mono) auto
qed (insert L fin, auto)
also have "\ = (INF i\I. emeasure M (F i))"
proof (intro antisym INF_greatest)
show "i \ I \ (INF i. emeasure M (F (L i))) \ emeasure M (F i)" for i
by (intro INF_lower2[of i]) auto
qed (insert L, auto intro: INF_lower)
also have "(\i. F (L i)) = (\i\I. F i)"
proof (intro antisym INF_greatest)
show "i \ I \ (\i. F (L i)) \ F i" for i
by (intro INF_lower2[of i]) auto
qed (insert L, auto)
finally show ?thesis .
qed
lemma Lim_emeasure_decseq:
assumes A: "range A \ sets M" "decseq A" and fin: "\i. emeasure M (A i) \ \"
shows "(\i. emeasure M (A i)) \ emeasure M (\i. A i)"
using LIMSEQ_INF[OF decseq_emeasure, OF A]
using INF_emeasure_decseq[OF A fin] by simp
lemma emeasure_lfp'[consumes 1, case_names cont measurable]:
assumes "P M"
assumes cont: "sup_continuous F"
assumes *: "\M A. P M \ (\N. P N \ Measurable.pred N A) \ Measurable.pred M (F A)"
shows "emeasure M {x\space M. lfp F x} = (SUP i. emeasure M {x\space M. (F ^^ i) (\x. False) x})"
proof -
have "emeasure M {x\space M. lfp F x} = emeasure M (\i. {x\space M. (F ^^ i) (\x. False) x})"
using sup_continuous_lfp[OF cont] by (auto simp add: bot_fun_def intro!: arg_cong2[where f=emeasure])
moreover { fix i from \<open>P M\<close> have "{x\<in>space M. (F ^^ i) (\<lambda>x. False) x} \<in> sets M"
by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) }
moreover have "incseq (\i. {x\space M. (F ^^ i) (\x. False) x})"
proof (rule incseq_SucI)
fix i
have "(F ^^ i) (\x. False) \ (F ^^ (Suc i)) (\x. False)"
proof (induct i)
case 0 show ?case by (simp add: le_fun_def)
next
case Suc thus ?case using monoD[OF sup_continuous_mono[OF cont] Suc] by auto
qed
then show "{x \ space M. (F ^^ i) (\x. False) x} \ {x \ space M. (F ^^ Suc i) (\x. False) x}"
by auto
qed
ultimately show ?thesis
by (subst SUP_emeasure_incseq) auto
qed
lemma emeasure_lfp:
assumes [simp]: "\s. sets (M s) = sets N"
assumes cont: "sup_continuous F" "sup_continuous f"
assumes meas: "\P. Measurable.pred N P \ Measurable.pred N (F P)"
assumes iter: "\P s. Measurable.pred N P \ P \ lfp F \ emeasure (M s) {x\space N. F P x} = f (\s. emeasure (M s) {x\space N. P x}) s"
shows "emeasure (M s) {x\space N. lfp F x} = lfp f s"
proof (subst lfp_transfer_bounded[where \<alpha>="\<lambda>F s. emeasure (M s) {x\<in>space N. F x}" and g=f and f=F and P="Measurable.pred N", symmetric])
fix C assume "incseq C" "\i. Measurable.pred N (C i)"
then show "(\s. emeasure (M s) {x \ space N. (SUP i. C i) x}) = (SUP i. (\s. emeasure (M s) {x \ space N. C i x}))"
unfolding SUP_apply[abs_def]
by (subst SUP_emeasure_incseq) (auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])
qed (auto simp add: iter le_fun_def SUP_apply[abs_def] intro!: meas cont)
lemma emeasure_subadditive_finite:
"finite I \ A ` I \ sets M \ emeasure M (\i\I. A i) \ (\i\I. emeasure M (A i))"
by (rule sets.subadditive[OF emeasure_positive emeasure_additive]) auto
lemma emeasure_subadditive:
"A \ sets M \ B \ sets M \ emeasure M (A \ B) \ emeasure M A + emeasure M B"
using emeasure_subadditive_finite[of "{True, False}" "\True \ A | False \ B" M] by simp
lemma emeasure_subadditive_countably:
assumes "range f \ sets M"
shows "emeasure M (\i. f i) \ (\i. emeasure M (f i))"
proof -
have "emeasure M (\i. f i) = emeasure M (\i. disjointed f i)"
unfolding UN_disjointed_eq ..
also have "\ = (\i. emeasure M (disjointed f i))"
using sets.range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]
by (simp add: disjoint_family_disjointed comp_def)
also have "\ \ (\i. emeasure M (f i))"
using sets.range_disjointed_sets[OF assms] assms
by (auto intro!: suminf_le emeasure_mono disjointed_subset)
finally show ?thesis .
qed
lemma emeasure_insert:
assumes sets: "{x} \ sets M" "A \ sets M" and "x \ A"
shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
proof -
have "{x} \ A = {}" using \x \ A\ by auto
from plus_emeasure[OF sets this] show ?thesis by simp
qed
lemma emeasure_insert_ne:
"A \ {} \ {x} \ sets M \ A \ sets M \ x \ A \ emeasure M (insert x A) = emeasure M {x} + emeasure M A"
by (rule emeasure_insert)
lemma emeasure_eq_sum_singleton:
assumes "finite S" "\x. x \ S \ {x} \ sets M"
shows "emeasure M S = (\x\S. emeasure M {x})"
using sum_emeasure[of "\x. {x}" S M] assms
by (auto simp: disjoint_family_on_def subset_eq)
lemma sum_emeasure_cover:
assumes "finite S" and "A \ sets M" and br_in_M: "B ` S \ sets M"
assumes A: "A \ (\i\S. B i)"
assumes disj: "disjoint_family_on B S"
shows "emeasure M A = (\i\S. emeasure M (A \ (B i)))"
proof -
have "(\i\S. emeasure M (A \ (B i))) = emeasure M (\i\S. A \ (B i))"
proof (rule sum_emeasure)
show "disjoint_family_on (\i. A \ B i) S"
using \<open>disjoint_family_on B S\<close>
unfolding disjoint_family_on_def by auto
qed (insert assms, auto)
also have "(\i\S. A \ (B i)) = A"
using A by auto
finally show ?thesis by simp
qed
lemma emeasure_eq_0:
"N \ sets M \ emeasure M N = 0 \ K \ N \ emeasure M K = 0"
by (metis emeasure_mono order_eq_iff zero_le)
lemma emeasure_UN_eq_0:
assumes "\i::nat. emeasure M (N i) = 0" and "range N \ sets M"
shows "emeasure M (\i. N i) = 0"
proof -
have "emeasure M (\i. N i) \ 0"
using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp
then show ?thesis
by (auto intro: antisym zero_le)
qed
lemma measure_eqI_finite:
assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"
assumes eq: "\a. a \ A \ emeasure M {a} = emeasure N {a}"
shows "M = N"
proof (rule measure_eqI)
fix X assume "X \ sets M"
then have X: "X \ A" by auto
then have "emeasure M X = (\a\X. emeasure M {a})"
using \<open>finite A\<close> by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset)
also have "\ = (\a\X. emeasure N {a})"
using X eq by (auto intro!: sum.cong)
also have "\ = emeasure N X"
using X \<open>finite A\<close> by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset)
finally show "emeasure M X = emeasure N X" .
qed simp
lemma measure_eqI_generator_eq:
fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \ 'a set"
assumes "Int_stable E" "E \ Pow \"
and eq: "\X. X \ E \ emeasure M X = emeasure N X"
and M: "sets M = sigma_sets \ E"
and N: "sets N = sigma_sets \ E"
and A: "range A \ E" "(\i. A i) = \" "\i. emeasure M (A i) \ \"
shows "M = N"
proof -
let ?\<mu> = "emeasure M" and ?\<nu> = "emeasure N"
interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact
have "space M = \"
using sets.top[of M] sets.space_closed[of M] S.top S.space_closed \<open>sets M = sigma_sets \<Omega> E\<close>
by blast
{ fix F D assume "F \ E" and "?\ F \ \"
then have [intro]: "F \ sigma_sets \ E" by auto
have "?\ F \ \" using \?\ F \ \\ \F \ E\ eq by simp
assume "D \ sets M"
with \<open>Int_stable E\<close> \<open>E \<subseteq> Pow \<Omega>\<close> have "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
unfolding M
proof (induct rule: sigma_sets_induct_disjoint)
case (basic A)
then have "F \ A \ E" using \Int_stable E\ \F \ E\ by (auto simp: Int_stable_def)
then show ?case using eq by auto
next
case empty then show ?case by simp
next
case (compl A)
then have **: "F \ (\ - A) = F - (F \ A)"
and [intro]: "F \ A \ sigma_sets \ E"
using \<open>F \<in> E\<close> S.sets_into_space by (auto simp: M)
have "?\ (F \ A) \ ?\ F" by (auto intro!: emeasure_mono simp: M N)
then have "?\ (F \ A) \ \" using \?\ F \ \\ by (auto simp: top_unique)
have "?\ (F \ A) \ ?\ F" by (auto intro!: emeasure_mono simp: M N)
then have "?\ (F \ A) \ \" using \?\ F \ \\ by (auto simp: top_unique)
then have "?\ (F \ (\ - A)) = ?\ F - ?\ (F \ A)" unfolding **
using \<open>F \<inter> A \<in> sigma_sets \<Omega> E\<close> by (auto intro!: emeasure_Diff simp: M N)
also have "\ = ?\ F - ?\ (F \ A)" using eq \F \ E\ compl by simp
also have "\ = ?\ (F \ (\ - A))" unfolding **
using \<open>F \<inter> A \<in> sigma_sets \<Omega> E\<close> \<open>?\<nu> (F \<inter> A) \<noteq> \<infinity>\<close>
by (auto intro!: emeasure_Diff[symmetric] simp: M N)
finally show ?case
using \<open>space M = \<Omega>\<close> by auto
next
case (union A)
then have "?\ (\x. F \ A x) = ?\ (\x. F \ A x)"
by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N)
with A show ?case
by auto
qed }
note * = this
show "M = N"
proof (rule measure_eqI)
show "sets M = sets N"
using M N by simp
have [simp, intro]: "\i. A i \ sets M"
using A(1) by (auto simp: subset_eq M)
fix F assume "F \ sets M"
let ?D = "disjointed (\i. F \ A i)"
from \<open>space M = \<Omega>\<close> have F_eq: "F = (\<Union>i. ?D i)"
using \<open>F \<in> sets M\<close>[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq)
have [simp, intro]: "\i. ?D i \ sets M"
using sets.range_disjointed_sets[of "\i. F \ A i" M] \F \ sets M\
by (auto simp: subset_eq)
have "disjoint_family ?D"
by (auto simp: disjoint_family_disjointed)
moreover
have "(\i. emeasure M (?D i)) = (\i. emeasure N (?D i))"
proof (intro arg_cong[where f=suminf] ext)
fix i
have "A i \ ?D i = ?D i"
by (auto simp: disjointed_def)
then show "emeasure M (?D i) = emeasure N (?D i)"
using *[of "A i" "?D i", OF _ A(3)] A(1) by auto
qed
ultimately show "emeasure M F = emeasure N F"
by (simp add: image_subset_iff \<open>sets M = sets N\<close>[symmetric] F_eq[symmetric] suminf_emeasure)
qed
qed
lemma space_empty: "space M = {} \ M = count_space {}"
by (rule measure_eqI) (simp_all add: space_empty_iff)
lemma measure_eqI_generator_eq_countable:
fixes M N :: "'a measure" and E :: "'a set set" and A :: "'a set set"
assumes E: "Int_stable E" "E \ Pow \" "\X. X \ E \ emeasure M X = emeasure N X"
and sets: "sets M = sigma_sets \ E" "sets N = sigma_sets \ E"
and A: "A \ E" "(\A) = \" "countable A" "\a. a \ A \ emeasure M a \ \"
shows "M = N"
proof cases
assume "\ = {}"
have *: "sigma_sets \ E = sets (sigma \ E)"
using E(2) by simp
have "space M = \" "space N = \"
using sets E(2) unfolding * by (auto dest: sets_eq_imp_space_eq simp del: sets_measure_of)
then show "M = N"
unfolding \<open>\<Omega> = {}\<close> by (auto dest: space_empty)
next
assume "\ \ {}" with \\A = \\ have "A \ {}" by auto
from this \<open>countable A\<close> have rng: "range (from_nat_into A) = A"
by (rule range_from_nat_into)
show "M = N"
proof (rule measure_eqI_generator_eq[OF E sets])
show "range (from_nat_into A) \ E"
unfolding rng using \<open>A \<subseteq> E\<close> .
show "(\i. from_nat_into A i) = \"
unfolding rng using \<open>\<Union>A = \<Omega>\<close> .
show "emeasure M (from_nat_into A i) \ \" for i
using rng by (intro A) auto
qed
qed
lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
proof (intro measure_eqI emeasure_measure_of_sigma)
show "sigma_algebra (space M) (sets M)" ..
show "positive (sets M) (emeasure M)"
by (simp add: positive_def)
show "countably_additive (sets M) (emeasure M)"
by (simp add: emeasure_countably_additive)
qed simp_all
subsection \<open>\<open>\<mu>\<close>-null sets\<close>
definition\<^marker>\<open>tag important\<close> null_sets :: "'a measure \<Rightarrow> 'a set set" where
"null_sets M = {N\sets M. emeasure M N = 0}"
lemma null_setsD1[dest]: "A \ null_sets M \ emeasure M A = 0"
by (simp add: null_sets_def)
lemma null_setsD2[dest]: "A \ null_sets M \ A \ sets M"
unfolding null_sets_def by simp
lemma null_setsI[intro]: "emeasure M A = 0 \ A \ sets M \ A \ null_sets M"
unfolding null_sets_def by simp
interpretation null_sets: ring_of_sets "space M" "null_sets M" for M
proof (rule ring_of_setsI)
show "null_sets M \ Pow (space M)"
using sets.sets_into_space by auto
show "{} \ null_sets M"
by auto
fix A B assume null_sets: "A \ null_sets M" "B \ null_sets M"
then have sets: "A \ sets M" "B \ sets M"
by auto
then have *: "emeasure M (A \ B) \ emeasure M A + emeasure M B"
"emeasure M (A - B) \ emeasure M A"
by (auto intro!: emeasure_subadditive emeasure_mono)
then have "emeasure M B = 0" "emeasure M A = 0"
using null_sets by auto
with sets * show "A - B \ null_sets M" "A \ B \ null_sets M"
by (auto intro!: antisym zero_le)
qed
lemma UN_from_nat_into:
assumes I: "countable I" "I \ {}"
shows "(\i\I. N i) = (\i. N (from_nat_into I i))"
proof -
have "(\i\I. N i) = \(N ` range (from_nat_into I))"
using I by simp
also have "\ = (\i. (N \ from_nat_into I) i)"
by simp
finally show ?thesis by simp
qed
lemma null_sets_UN':
assumes "countable I"
assumes "\i. i \ I \ N i \ null_sets M"
shows "(\i\I. N i) \ null_sets M"
proof cases
assume "I = {}" then show ?thesis by simp
next
assume "I \ {}"
show ?thesis
proof (intro conjI CollectI null_setsI)
show "(\i\I. N i) \ sets M"
using assms by (intro sets.countable_UN') auto
have "emeasure M (\i\I. N i) \ (\n. emeasure M (N (from_nat_into I n)))"
unfolding UN_from_nat_into[OF \<open>countable I\<close> \<open>I \<noteq> {}\<close>]
using assms \<open>I \<noteq> {}\<close> by (intro emeasure_subadditive_countably) (auto intro: from_nat_into)
also have "(\n. emeasure M (N (from_nat_into I n))) = (\_. 0)"
using assms \<open>I \<noteq> {}\<close> by (auto intro: from_nat_into)
finally show "emeasure M (\i\I. N i) = 0"
by (intro antisym zero_le) simp
qed
qed
lemma null_sets_UN[intro]:
"(\i::'i::countable. N i \ null_sets M) \ (\i. N i) \ null_sets M"
by (rule null_sets_UN') auto
lemma null_set_Int1:
assumes "B \ null_sets M" "A \ sets M" shows "A \ B \ null_sets M"
proof (intro CollectI conjI null_setsI)
show "emeasure M (A \ B) = 0" using assms
by (intro emeasure_eq_0[of B _ "A \ B"]) auto
qed (insert assms, auto)
lemma null_set_Int2:
assumes "B \ null_sets M" "A \ sets M" shows "B \ A \ null_sets M"
using assms by (subst Int_commute) (rule null_set_Int1)
lemma emeasure_Diff_null_set:
assumes "B \ null_sets M" "A \ sets M"
shows "emeasure M (A - B) = emeasure M A"
proof -
have *: "A - B = (A - (A \ B))" by auto
have "A \ B \ null_sets M" using assms by (rule null_set_Int1)
then show ?thesis
unfolding * using assms
by (subst emeasure_Diff) auto
qed
lemma null_set_Diff:
assumes "B \ null_sets M" "A \ sets M" shows "B - A \ null_sets M"
proof (intro CollectI conjI null_setsI)
show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
qed (insert assms, auto)
lemma emeasure_Un_null_set:
assumes "A \ sets M" "B \ null_sets M"
shows "emeasure M (A \ B) = emeasure M A"
proof -
have *: "A \ B = A \ (B - A)" by auto
have "B - A \ null_sets M" using assms(2,1) by (rule null_set_Diff)
then show ?thesis
unfolding * using assms
by (subst plus_emeasure[symmetric]) auto
qed
lemma emeasure_Un':
assumes "A \ sets M" "B \ sets M" "A \ B \ null_sets M"
shows "emeasure M (A \ B) = emeasure M A + emeasure M B"
proof -
have "A \ B = A \ (B - A \ B)" by blast
also have "emeasure M \ = emeasure M A + emeasure M (B - A \ B)"
using assms by (subst plus_emeasure) auto
also have "emeasure M (B - A \ B) = emeasure M B"
using assms by (intro emeasure_Diff_null_set) auto
finally show ?thesis .
qed
subsection \<open>The almost everywhere filter (i.e.\ quantifier)\<close>
definition\<^marker>\<open>tag important\<close> ae_filter :: "'a measure \<Rightarrow> 'a filter" where
"ae_filter M = (INF N\null_sets M. principal (space M - N))"
abbreviation almost_everywhere :: "'a measure \ ('a \ bool) \ bool" where
"almost_everywhere M P \ eventually P (ae_filter M)"
syntax
"_almost_everywhere" :: "pttrn \ 'a \ bool \ bool" ("AE _ in _. _" [0,0,10] 10)
translations
"AE x in M. P" \<rightleftharpoons> "CONST almost_everywhere M (\<lambda>x. P)"
abbreviation
"set_almost_everywhere A M P \ AE x in M. x \ A \ P x"
syntax
"_set_almost_everywhere" :: "pttrn \ 'a set \ 'a \ bool \ bool"
("AE _\_ in _./ _" [0,0,0,10] 10)
translations
"AE x\A in M. P" \ "CONST set_almost_everywhere A M (\x. P)"
lemma eventually_ae_filter: "eventually P (ae_filter M) \ (\N\null_sets M. {x \ space M. \ P x} \ N)"
unfolding ae_filter_def by (subst eventually_INF_base) (auto simp: eventually_principal subset_eq)
lemma AE_I':
"N \ null_sets M \ {x\space M. \ P x} \ N \ (AE x in M. P x)"
unfolding eventually_ae_filter by auto
lemma AE_iff_null:
assumes "{x\space M. \ P x} \ sets M" (is "?P \ sets M")
shows "(AE x in M. P x) \ {x\space M. \ P x} \ null_sets M"
proof
assume "AE x in M. P x" then obtain N where N: "N \ sets M" "?P \ N" "emeasure M N = 0"
unfolding eventually_ae_filter by auto
have "emeasure M ?P \ emeasure M N"
using assms N(1,2) by (auto intro: emeasure_mono)
then have "emeasure M ?P = 0"
unfolding \<open>emeasure M N = 0\<close> by auto
then show "?P \ null_sets M" using assms by auto
next
assume "?P \ null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
qed
lemma AE_iff_null_sets:
"N \ sets M \ N \ null_sets M \ (AE x in M. x \ N)"
using Int_absorb1[OF sets.sets_into_space, of N M]
by (subst AE_iff_null) (auto simp: Int_def[symmetric])
lemma AE_not_in:
"N \ null_sets M \ AE x in M. x \ N"
by (metis AE_iff_null_sets null_setsD2)
lemma AE_iff_measurable:
"N \ sets M \ {x\space M. \ P x} = N \ (AE x in M. P x) \ emeasure M N = 0"
using AE_iff_null[of _ P] by auto
lemma AE_E[consumes 1]:
assumes "AE x in M. P x"
obtains N where "{x \ space M. \ P x} \ N" "emeasure M N = 0" "N \ sets M"
using assms unfolding eventually_ae_filter by auto
lemma AE_E2:
assumes "AE x in M. P x" "{x\space M. P x} \ sets M"
shows "emeasure M {x\space M. \ P x} = 0" (is "emeasure M ?P = 0")
proof -
have "{x\space M. \ P x} = space M - {x\space M. P x}" by auto
with AE_iff_null[of M P] assms show ?thesis by auto
qed
lemma AE_E3:
assumes "AE x in M. P x"
obtains N where "\x. x \ space M - N \ P x" "N \ null_sets M"
using assms unfolding eventually_ae_filter by auto
lemma AE_I:
assumes "{x \ space M. \ P x} \ N" "emeasure M N = 0" "N \ sets M"
shows "AE x in M. P x"
using assms unfolding eventually_ae_filter by auto
lemma AE_mp[elim!]:
assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \ Q x"
shows "AE x in M. Q x"
proof -
from AE_P obtain A where P: "{x\space M. \ P x} \ A"
and A: "A \ sets M" "emeasure M A = 0"
by (auto elim!: AE_E)
from AE_imp obtain B where imp: "{x\space M. P x \ \ Q x} \ B"
and B: "B \ sets M" "emeasure M B = 0"
by (auto elim!: AE_E)
show ?thesis
proof (intro AE_I)
have "emeasure M (A \ B) \ 0"
using emeasure_subadditive[of A M B] A B by auto
then show "A \ B \ sets M" "emeasure M (A \ B) = 0"
using A B by auto
show "{x\space M. \ Q x} \ A \ B"
using P imp by auto
qed
qed
text \<open>The next lemma is convenient to combine with a lemma whose conclusion is of the
form \<open>AE x in M. P x = Q x\<close>: for such a lemma, there is no \<open>[symmetric]\<close> variant,
but using \<open>AE_symmetric[OF...]\<close> will replace it.\<close>
(* depricated replace by laws about eventually *)
lemma
shows AE_iffI: "AE x in M. P x \ AE x in M. P x \ Q x \ AE x in M. Q x"
and AE_disjI1: "AE x in M. P x \ AE x in M. P x \ Q x"
and AE_disjI2: "AE x in M. Q x \ AE x in M. P x \ Q x"
and AE_conjI: "AE x in M. P x \ AE x in M. Q x \ AE x in M. P x \ Q x"
and AE_conj_iff[simp]: "(AE x in M. P x \ Q x) \ (AE x in M. P x) \ (AE x in M. Q x)"
by auto
lemma AE_symmetric:
assumes "AE x in M. P x = Q x"
shows "AE x in M. Q x = P x"
using assms by auto
lemma AE_impI:
"(P \ AE x in M. Q x) \ AE x in M. P \ Q x"
by fastforce
lemma AE_measure:
assumes AE: "AE x in M. P x" and sets: "{x\space M. P x} \ sets M" (is "?P \ sets M")
shows "emeasure M {x\space M. P x} = emeasure M (space M)"
proof -
from AE_E[OF AE] guess N . note N = this
with sets have "emeasure M (space M) \ emeasure M (?P \ N)"
by (intro emeasure_mono) auto
also have "\ \ emeasure M ?P + emeasure M N"
using sets N by (intro emeasure_subadditive) auto
also have "\ = emeasure M ?P" using N by simp
finally show "emeasure M ?P = emeasure M (space M)"
using emeasure_space[of M "?P"] by auto
qed
lemma AE_space: "AE x in M. x \ space M"
by (rule AE_I[where N="{}"]) auto
lemma AE_I2[simp, intro]:
"(\x. x \ space M \ P x) \ AE x in M. P x"
using AE_space by force
lemma AE_Ball_mp:
"\x\space M. P x \ AE x in M. P x \ Q x \ AE x in M. Q x"
by auto
lemma AE_cong[cong]:
"(\x. x \ space M \ P x \ Q x) \ (AE x in M. P x) \ (AE x in M. Q x)"
by auto
lemma AE_cong_simp: "M = N \ (\x. x \ space N =simp=> P x = Q x) \ (AE x in M. P x) \ (AE x in N. Q x)"
by (auto simp: simp_implies_def)
lemma AE_all_countable:
"(AE x in M. \i. P i x) \ (\i::'i::countable. AE x in M. P i x)"
proof
assume "\i. AE x in M. P i x"
from this[unfolded eventually_ae_filter Bex_def, THEN choice]
obtain N where N: "\i. N i \ null_sets M" "\i. {x\space M. \ P i x} \ N i" by auto
have "{x\space M. \ (\i. P i x)} \ (\i. {x\space M. \ P i x})" by auto
also have "\ \ (\i. N i)" using N by auto
finally have "{x\space M. \ (\i. P i x)} \ (\i. N i)" .
moreover from N have "(\i. N i) \ null_sets M"
by (intro null_sets_UN) auto
ultimately show "AE x in M. \i. P i x"
unfolding eventually_ae_filter by auto
qed auto
lemma AE_ball_countable:
assumes [intro]: "countable X"
shows "(AE x in M. \y\X. P x y) \ (\y\X. AE x in M. P x y)"
proof
assume "\y\X. AE x in M. P x y"
from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]
obtain N where N: "\y. y \ X \ N y \ null_sets M" "\y. y \ X \ {x\space M. \ P x y} \ N y"
by auto
have "{x\space M. \ (\y\X. P x y)} \ (\y\X. {x\space M. \ P x y})"
by auto
also have "\ \ (\y\X. N y)"
using N by auto
finally have "{x\space M. \ (\y\X. P x y)} \ (\y\X. N y)" .
moreover from N have "(\y\X. N y) \ null_sets M"
by (intro null_sets_UN') auto
ultimately show "AE x in M. \y\X. P x y"
unfolding eventually_ae_filter by auto
qed auto
lemma AE_ball_countable':
"(\N. N \ I \ AE x in M. P N x) \ countable I \ AE x in M. \N \ I. P N x"
unfolding AE_ball_countable by simp
lemma AE_pairwise: "countable F \ pairwise (\A B. AE x in M. R x A B) F \ (AE x in M. pairwise (R x) F)"
unfolding pairwise_alt by (simp add: AE_ball_countable)
lemma AE_discrete_difference:
assumes X: "countable X"
assumes null: "\x. x \ X \ emeasure M {x} = 0"
assumes sets: "\x. x \ X \ {x} \ sets M"
shows "AE x in M. x \ X"
proof -
have "(\x\X. {x}) \ null_sets M"
using assms by (intro null_sets_UN') auto
from AE_not_in[OF this] show "AE x in M. x \ X"
by auto
qed
lemma AE_finite_all:
assumes f: "finite S" shows "(AE x in M. \i\S. P i x) \ (\i\S. AE x in M. P i x)"
using f by induct auto
lemma AE_finite_allI:
assumes "finite S"
shows "(\s. s \ S \ AE x in M. Q s x) \ AE x in M. \s\S. Q s x"
using AE_finite_all[OF \<open>finite S\<close>] by auto
lemma emeasure_mono_AE:
assumes imp: "AE x in M. x \ A \ x \ B"
and B: "B \ sets M"
shows "emeasure M A \ emeasure M B"
proof cases
assume A: "A \ sets M"
from imp obtain N where N: "{x\space M. \ (x \ A \ x \ B)} \ N" "N \ null_sets M"
by (auto simp: eventually_ae_filter)
have "emeasure M A = emeasure M (A - N)"
using N A by (subst emeasure_Diff_null_set) auto
also have "emeasure M (A - N) \ emeasure M (B - N)"
using N A B sets.sets_into_space by (auto intro!: emeasure_mono)
also have "emeasure M (B - N) = emeasure M B"
using N B by (subst emeasure_Diff_null_set) auto
finally show ?thesis .
qed (simp add: emeasure_notin_sets)
lemma emeasure_eq_AE:
assumes iff: "AE x in M. x \ A \ x \ B"
assumes A: "A \ sets M" and B: "B \ sets M"
shows "emeasure M A = emeasure M B"
using assms by (safe intro!: antisym emeasure_mono_AE) auto
lemma emeasure_Collect_eq_AE:
"AE x in M. P x \ Q x \ Measurable.pred M Q \ Measurable.pred M P \
emeasure M {x\<in>space M. P x} = emeasure M {x\<in>space M. Q x}"
by (intro emeasure_eq_AE) auto
lemma emeasure_eq_0_AE: "AE x in M. \ P x \ emeasure M {x\space M. P x} = 0"
using AE_iff_measurable[OF _ refl, of M "\x. \ P x"]
by (cases "{x\space M. P x} \ sets M") (simp_all add: emeasure_notin_sets)
lemma emeasure_0_AE:
assumes "emeasure M (space M) = 0"
shows "AE x in M. P x"
using eventually_ae_filter assms by blast
lemma emeasure_add_AE:
assumes [measurable]: "A \ sets M" "B \ sets M" "C \ sets M"
assumes 1: "AE x in M. x \ C \ x \ A \ x \ B"
assumes 2: "AE x in M. \ (x \ A \ x \ B)"
shows "emeasure M C = emeasure M A + emeasure M B"
proof -
have "emeasure M C = emeasure M (A \ B)"
by (rule emeasure_eq_AE) (insert 1, auto)
also have "\ = emeasure M A + emeasure M (B - A)"
by (subst plus_emeasure) auto
also have "emeasure M (B - A) = emeasure M B"
by (rule emeasure_eq_AE) (insert 2, auto)
finally show ?thesis .
qed
subsection \<open>\<open>\<sigma>\<close>-finite Measures\<close>
locale\<^marker>\<open>tag important\<close> sigma_finite_measure =
fixes M :: "'a measure"
assumes sigma_finite_countable:
"\A::'a set set. countable A \ A \ sets M \ (\A) = space M \ (\a\A. emeasure M a \ \)"
lemma (in sigma_finite_measure) sigma_finite:
obtains A :: "nat \ 'a set"
where "range A \ sets M" "(\i. A i) = space M" "\i. emeasure M (A i) \ \"
proof -
obtain A :: "'a set set" where
[simp]: "countable A" and
A: "A \ sets M" "(\A) = space M" "\a. a \ A \ emeasure M a \ \"
using sigma_finite_countable by metis
show thesis
proof cases
assume "A = {}" with \<open>(\<Union>A) = space M\<close> show thesis
by (intro that[of "\_. {}"]) auto
next
assume "A \ {}"
show thesis
proof
show "range (from_nat_into A) \ sets M"
using \<open>A \<noteq> {}\<close> A by auto
have "(\i. from_nat_into A i) = \A"
using range_from_nat_into[OF \<open>A \<noteq> {}\<close> \<open>countable A\<close>] by auto
with A show "(\i. from_nat_into A i) = space M"
by auto
qed (intro A from_nat_into \<open>A \<noteq> {}\<close>)
qed
qed
lemma (in sigma_finite_measure) sigma_finite_disjoint:
obtains A :: "nat \ 'a set"
where "range A \ sets M" "(\i. A i) = space M" "\i. emeasure M (A i) \ \" "disjoint_family A"
proof -
obtain A :: "nat \ 'a set" where
range: "range A \ sets M" and
space: "(\i. A i) = space M" and
measure: "\i. emeasure M (A i) \ \"
using sigma_finite by blast
show thesis
proof (rule that[of "disjointed A"])
show "range (disjointed A) \ sets M"
by (rule sets.range_disjointed_sets[OF range])
show "(\i. disjointed A i) = space M"
and "disjoint_family (disjointed A)"
using disjoint_family_disjointed UN_disjointed_eq[of A] space range
by auto
show "emeasure M (disjointed A i) \ \" for i
proof -
have "emeasure M (disjointed A i) \ emeasure M (A i)"
using range disjointed_subset[of A i] by (auto intro!: emeasure_mono)
then show ?thesis using measure[of i] by (auto simp: top_unique)
qed
qed
qed
lemma (in sigma_finite_measure) sigma_finite_incseq:
obtains A :: "nat \ 'a set"
where "range A \ sets M" "(\i. A i) = space M" "\i. emeasure M (A i) \ \" "incseq A"
proof -
obtain F :: "nat \ 'a set" where
F: "range F \ sets M" "(\i. F i) = space M" "\i. emeasure M (F i) \ \"
using sigma_finite by blast
show thesis
proof (rule that[of "\n. \i\n. F i"])
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